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$\dfrac{ \sqrt{ 3 } }{ 2+ \sqrt{ 6 } } - \dfrac{ \sqrt{ 3 } }{ 2- \sqrt{ 6 } }$
$3 \sqrt{ 2 }$
Calculate the value
$\dfrac { \sqrt{ 3 } } { 2 + \sqrt{ 6 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Find the conjugate irrational number of denominator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ 3 } } { 2 + \sqrt{ 6 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 - \sqrt{ 6 } } { 2 - \sqrt{ 6 } } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ 3 } } { 2 + \sqrt{ 6 } } \times \dfrac { 2 - \sqrt{ 6 } } { 2 - \sqrt{ 6 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$\color{#FF6800}{ \dfrac { \sqrt{ 3 } \left ( 2 - \sqrt{ 6 } \right ) } { \left ( 2 + \sqrt{ 6 } \right ) \left ( 2 - \sqrt{ 6 } \right ) } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } \right ) } { \left ( 2 + \sqrt{ 6 } \right ) \left ( 2 - \sqrt{ 6 } \right ) } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Multiply each term in parentheses by $ \sqrt{ 3 }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } \right ) } { \left ( 2 + \sqrt{ 6 } \right ) \left ( 2 - \sqrt{ 6 } \right ) } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } \right ) } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 6 } \right ) ^ { 2 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 6 } \right ) ^ { 2 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { 4 - \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Calculate power $ $
$\dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { 4 - \color{#FF6800}{ 6 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Simplify the expression $ $
$\dfrac { \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + \sqrt{ 3 } \times \left ( - \sqrt{ 6 } \right ) } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { 2 \sqrt{ 3 } + \sqrt{ 3 } \times \left ( \color{#FF6800}{ - } \sqrt{ 6 } \right ) } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Move the (-) sign forward $ $
$\dfrac { 2 \sqrt{ 3 } \color{#FF6800}{ - } \sqrt{ 3 } \sqrt{ 6 } } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { 2 \sqrt{ 3 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 6 } } } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Calculate multiplication $ $
$\dfrac { 2 \sqrt{ 3 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { 4 - 6 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { 2 \sqrt{ 3 } - \left ( 3 \sqrt{ 2 } \right ) } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 6 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Subtract $ 6 $ from $ 4$
$\dfrac { 2 \sqrt{ 3 } - \left ( 3 \sqrt{ 2 } \right ) } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { 2 \sqrt{ 3 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) } { - 2 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Get rid of unnecessary parentheses $ $
$\dfrac { 2 \sqrt{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } } { - 2 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$\dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Move the minus sign to the front of the fraction $ $
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } }$
$ $ Find the conjugate irrational number of denominator $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \left ( \color{#FF6800}{ \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } } } \color{#FF6800}{ \times } \color{#FF6800}{ \dfrac { 2 + \sqrt{ 6 } } { 2 + \sqrt{ 6 } } } \right )$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \left ( \dfrac { \sqrt{ 3 } } { 2 - \sqrt{ 6 } } \times \dfrac { 2 + \sqrt{ 6 } } { 2 + \sqrt{ 6 } } \right )$
$ $ The denominator is multiplied by denominator, and the numerator is multiplied by numerator $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \color{#FF6800}{ \dfrac { \sqrt{ 3 } \left ( 2 + \sqrt{ 6 } \right ) } { \left ( 2 - \sqrt{ 6 } \right ) \left ( 2 + \sqrt{ 6 } \right ) } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \right ) } { \left ( 2 - \sqrt{ 6 } \right ) \left ( 2 + \sqrt{ 6 } \right ) }$
$ $ Multiply each term in parentheses by $ \sqrt{ 3 }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 6 } } } { \left ( 2 - \sqrt{ 6 } \right ) \left ( 2 + \sqrt{ 6 } \right ) }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \sqrt{ 6 } } { \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 6 } } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 6 } } \right ) }$
$ $ Expand the expression using $ \left(a - b\right)\left(a + b\right) = a^{2} - b^{2}$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 } \sqrt{ 6 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ \color{#FF6800}{ 3 } } \sqrt{ \color{#FF6800}{ 6 } } } { 2 ^ { 2 } - \left ( \sqrt{ 6 } \right ) ^ { 2 } }$
$ $ Arrange the expression $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } } } { 2 ^ { 2 } - \left ( \sqrt{ 6 } \right ) ^ { 2 } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 \times 6 } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - \left ( \sqrt{ 6 } \right ) ^ { 2 } }$
$ $ Calculate power $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 \times 6 } } { \color{#FF6800}{ 4 } - \left ( \sqrt{ 6 } \right ) ^ { 2 } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 \times 6 } } { 4 - \left ( \sqrt{ \color{#FF6800}{ 6 } } \right ) ^ { \color{#FF6800}{ 2 } } }$
$ $ Calculate power $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ 3 } \times 2 + \sqrt{ 3 \times 6 } } { 4 - \color{#FF6800}{ 6 } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } + \sqrt{ 3 \times 6 } } { 4 - 6 }$
$ $ Simplify the expression $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + \sqrt{ 3 \times 6 } } { 4 - 6 }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \sqrt{ \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 } } } { 4 - 6 }$
$ $ Multiply $ 3 $ and $ 6$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \sqrt{ \color{#FF6800}{ 18 } } } { 4 - 6 }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \sqrt{ 18 } } { \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 6 } }$
$ $ Subtract $ 6 $ from $ 4$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \sqrt{ 18 } } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \sqrt{ \color{#FF6800}{ 18 } } } { - 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } } { - 2 }$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \dfrac { 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { \color{#FF6800}{ - } \color{#FF6800}{ 2 } }$
$ $ Move the minus sign to the front of the fraction $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } - \left ( \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 } } \right )$
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \dfrac { 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 } \right )$
$ $ Simplify Minus $ $
$- \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } + \dfrac { 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 \sqrt{ 3 } - 3 \sqrt{ 2 } } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 } }$
$ $ Combine the fraction with the same denominator $ $
$\dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } } { 2 }$
$\dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \right ) + 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 }$
$ $ Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses $ $
$\dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } + 2 \sqrt{ 3 } + 3 \sqrt{ 2 } } { 2 }$
$\dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + 3 \sqrt{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } + 3 \sqrt{ 2 } } { 2 }$
$ $ Eliminate opponent number $ $
$\dfrac { 3 \sqrt{ 2 } + 3 \sqrt{ 2 } } { 2 }$
$\dfrac { \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } } } { 2 }$
$ $ Calculate between similar terms $ $
$\dfrac { \color{#FF6800}{ 6 } \sqrt{ \color{#FF6800}{ 2 } } } { 2 }$
$\color{#FF6800}{ \dfrac { 6 \sqrt{ 2 } } { 2 } }$
$ $ Reduce the fraction $ $
$\color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 2 } }$
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